Question

In the following exercises, evaluate each determinant by expanding by minors along the first row.

$\left|\begin{array}{ccc}4& -1& -2\\ -3& -2& 1\\ -2& -5& 7\end{array}\right|$

Short answer

The value of the determinant is$-77.$

Step-by-step solution

Step 1. Given Information

The given matrix is

$\left|\begin{array}{ccc}4& -1& -2\\ -3& -2& 1\\ -2& -5& 7\end{array}\right|$

We have to evaluate the determinant by expanding by minors along the first row.

Step 2. Expand by minors along the first row

The first row is $\left|\begin{array}{ccc}4& -1& -2\end{array}\right|$

Now, let's expand by minors along the first row

$=4\left|\begin{array}{cc}-2& 1\\ -5& 7\end{array}\right|-(-1)\left|\begin{array}{cc}-3& 1\\ -2& 7\end{array}\right|-2\left|\begin{array}{cc}-3& -2\\ -2& -5\end{array}\right|$

Step 3. Evaluate the determinant

By expanding by minors along the first row we get,

$$\begin{gathered} =4\left|\begin{array}{cc}-2& 1\\ -5& 7\end{array}\right|-(-1)\left|\begin{array}{cc}-3& 1\\ -2& 7\end{array}\right|-2\left|\begin{array}{cc}-3& -2\\ -2& -5\end{array}\right| \\ =4\left[\right(-2\left)\right(7)-(1\left)\right(-5\left)\right]+1\left[\right(-3\left)\right(7)-(-2\left)\right(1\left)\right]-2\left[\right(-3\left)\right(-5)-(-2\left)\right(-2\left)\right] \\ =4[-9]+1[-19]-2\left[11\right] \\ =-77 \end{gathered}$$

Thus, the determinant is$-77.$